Because \(A = k\big [x,x^{-1}\big ] \cong k[x,y]/(xy-\mathbf {1})\)  admits a finite presentation
by two generators \(x\)  and \(y\) , the corresponding module of Kähler 1-forms
is isomorphic to the quotient \(\Omega ^1(A) \cong (A\,\mathrm {d}x \oplus A\,\mathrm {d}y) /\langle y\,\mathrm {d}x + x\,\mathrm {d}y\rangle \) ,
see, e.g., [1]}. The generator \(\mathrm {d}y\)  can be eliminated
by the relation \(\mathrm {d}y = -y^{2} \mathrm {d}x\) , which implies that \(\Omega ^1(A) = A\,\mathrm {d}x\) . (Note
that \(\mathrm {d}x^{-1} = -x^{-2} \mathrm {d}x\)  in this calculus.)
Restricting the Kähler 1-forms \(\Omega ^1(A)\)  to \(B= k\big [x^p,x^{-p}\big ]\subseteq k\big [x,x^{-1}\big ] = A\) 
defines the trivial differential calculus \(\Omega ^1(B) = 0\) . This follows as \(k\)  is
a field of characteristic \(p\) , by hypothesis, and hence \(\mathrm {d}(x^p) = p\, x^{p-1}\,\mathrm {d}x =0\) . On the other hand, the Kähler differential calculus on \(B\)  is non-trivial.We
are grateful to Christian Lomp for his comment, which led us to Example REF .

